Question REF  was also asked in [1]}, and in a slightly different form in [2]}.
While we do not answer Question REF  in the present article, we answer the closely related Questions REF  and REF , providing support for a negative answer to Question REF . Most techniques for analyzing the difference set \(A-A\)  of a set \(A\)  having positive upper Banach density actually provide information about the \(c\) -popular differences \(P_{c}(A):=\lbrace n: d^{*}(A\cap (A-n))>c\rbrace \) , where \(c>0\)  is some fixed number. Note that \(P_{c}(A)\subseteq A-A\)  for each \(c\ge 0\) . The following questions now present themselves.
